MiningMath

MiningMath

Loading...

Global Optimization with no stepwise process!

Azimuth Rotation Procedure

Estimated reading: 2 minutes 96 views

MiningMath supports the use of block models that have been rotated using an Azimuth rotation.

Rotation steps

This procedure performs a rotation of a point around a specific origin in the \(XY\) plane, while preserving the \(Z\) coordinate. Below are the steps involved, along with the corresponding mathematical formulas.

  1. Azimuth normalization

    First, the azimuth value is adjusted to ensure it lies within the range of \(0\) to \(360\) degrees. This is done by taking the remainder of the azimuth divided by \(360\)

    \(\text{azimuth} = \text{fmod}(\text{azimuth}, 360.0)\)

    In mathematics, the function \(\text{fmod}\) (floating-point modulus) computes the remainder of the division of two floating-point numbers. If the azimuth value is negative, \(360\) degrees is added to make it positive:

    \(\text{azimuth} = \begin{cases}
    \text{azimuth} + 360.0 & \text{if } \text{azimuth} < 0.0 \\
    \text{azimuth} & \text{otherwise}
    \end{cases}\)

  2. Inversion of Rotation Direction

    The adjusted azimuth is then inverted to perform the rotation in the opposite direction:

    \(\text{azimuth} = -\text{azimuth}\)

  3. Conversion from Degrees to Radians

    The azimuth in degrees is converted to radians, as trigonometric functions use radians:

    \(\text{radians} = \frac{\text{azimuth} \times \pi}{180.0}\)

  4. Calculation of New Coordinates

    The new \(x\) and \(y\) coordinates are calculated by applying the rotation in the \(XY\) plane around an origin \((x_0, y_0)\). The formulas used are:

    \(
    \text{px} = \left( (x - x_0) \times \cos(\text{radians}) + (y - y_0) \times \sin(\text{radians}) \right) + x_0\)

    \(
    \text{py} = \left( -(x - x_0) \times \sin(\text{radians}) + (y - y_0) \times \cos(\text{radians}) \right) + y_0\)

    Where \((x, y)\) are the initial coordinates of the point, and \((x_0, y_0)\) are the coordinates of the origin.

  5. Preservation of Z Coordinate

    The \(z\) coordinate of the original point remains unchanged during the rotation:

    \(z_{\text{final}} = z_{\text{initial}}\)

  6. Final Result

    The result of the procedure is a new point with coordinates \((p_x, p_y, z)\), where \(p_x\) and \(p_y\) are the new coordinates in the \(XY\) plane after the rotation, and \(z\) is the original coordinate in the \(Z\) axis.

    \(\text{New Point} = \{ p_x, p_y, z \}\)

Share this Doc

Azimuth Rotation Procedure

Or copy link

CONTENTS
Chat Icon

Hi, it's Mima here 😇 Ask me any questions!